Hooray, I finally did this mission! I don't know if it was the guides that did it for me or if they just gave me the confidence to try again. Anyway, here's what I did.
First of all, I'm not sure why some people say you should never let up on the gas ever. Letting up on the gas was critical to my success. I have tried various methods but whenever I hold the gas down 100% of the time, I always get too far ahead of the pack after the first turn. I held down the gas almost the entire time, but I slowed down just a tiny bit after I initially passed everyone to ensure they wouldn't fall too far behind. That was the only time I let up. After that, I made sure they were drafting me on the straights and the second half of each turn. By the beginning of the second lap, I could already see the yellow Skyline off in the distance. I repeated the same patter over and over again (except I didn't let up on the throttle in the turns; that was only on the very first turn in the first lap) until I got close. I even accidentally hit one of the other Skylines as I attempted to pass and suffered a 5 second penalty, but by then I was within drafting distance of the yellow Skyline so I just got back up to speed and passed him on the next to last straight.
I don't know how it's possible to win without somehow slowing down at least a tiny bit at some point to let the others catch up. I think that going into the first turn, the other cars are not quite moving as fast as they are in the later turns because they haven't had the chance to build up enough speed or draft you yet, so you have to help them draft you. I got a feel for how close they need to be by taking into account how far behind I was at the beginning (since at the very beginning of the mission you're basically at the very edge of the drafting zone).